Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 078, 22 pages      arXiv:0904.0565      https://doi.org/10.3842/SIGMA.2009.078
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

On Spinor Varieties and Their Secants

Laurent Manivel
Institut Fourier, Université de Grenoble I et CNRS, BP 74, 38402 Saint-Martin d'Hères, France

Received April 03, 2009, in final form July 21, 2009; Published online July 24, 2009

Abstract
We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type Dn, cubic equations exist if and only if n ≥ 9. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.

Key words: spinor variety; spin representation; secant variety; Freudenthal variety.

pdf (327 kb)   ps (235 kb)   tex (25 kb)

References

  1. Atiyah M.F., Bott R., A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451-491.
  2. Chen Y.M., Garsia A.M., Remmel J., Algorithms for plethysm, in Combinatorics and Algebra (Boulder, Colo., 1983), Contemp. Math., Vol. 34, Amer. Math. Soc., Providence, RI, 1984, 109-153.
  3. Chevalley C., The algebraic theory of spinors and Clifford algebras, Collected works, Vol. 2, Springer-Verlag, Berlin, 1997.
  4. Dress A.W.M., Wenzel W., A simple proof of an identity concerning Pfaffians of skew symmetric matrices, Adv. Math. 112 (1995), 120-134.
  5. Howe R., Tan E.C., Willenbring J.F., Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), 1601-1626, math.RT/0311159.
  6. Iarrobino A., Kanev V., Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, Vol. 1721, Springer-Verlag, Berlin, 1999.
  7. Kaji H., Yasukura O., Projective geometry of Freudenthal's varieties of certain type, Michigan Math. J. 52 (2004), 515-542.
  8. Kanev V., Chordal varieties of Veronese varieties and catalecticant matrices, J. Math. Sci. (New York) 94 (1999), 1114-1125, math.AG/9804141.
  9. Knuth D.E., Overlapping Pfaffians, Electron. J. Combin. 3 (1996), no. 2, paper 5, 13 pages, math.CO/9503234.
  10. Landsberg J.M., Manivel L., The projective geometry of Freudenthal's magic square, J. Algebra 239 (2001), 477-512, math.AG/9908039.
  11. Landsberg J.M., Manivel L., On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (2004), 397-422, math.AG/0311388.
  12. Landsberg J.M., Weyman J., On tangential varieties of rational homogeneous varieties, J. Lond. Math. Soc. (2) 76 (2007), 513-530, math.AG/0509388.
  13. Landsberg J.M., Weyman J., On secant varieties of compact Hermitian symmetric spaces, arXiv:0802.3402.
  14. LiE, A computer algebra package for Lie group computations, available at http://young.sp2mi.univ-poitiers.fr/~marc/LiE/.
  15. Zak F.L., Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, Vol. 127, American Mathematical Society, Providence, RI, 1993.

Previous article   Next article   Contents of Volume 5 (2009)